They don't. Although the electrical energy itself travels very quickly--some large fraction of the speed of light in most types of insulated wiring, nearly c in uninsulated wires, the electrons themselves travel very, very slowly. The speed at which they move is dependent primarly upon current and conductor cross section and is called the "drift velocity." In typical household loads and wiring, the drift velocity can be on the order of a few mm (yes, I mean millimeters) per second. In AC, they more or less just oscillate back and forth in place at the line frequency and don't really get anywhere at all.
Electrical conduction at the at atomic level in real materials is VERY tricky to understand and requires quite sophisticated quantum mechanics in order to get the full picture.
In most cases we use simplified models (such as the Drude model) which usually work quite well but there is no getting round the fact that you need to know a lot about solid state physics if you want to understand what is "really" going on.